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Lab #1: Motor Torque EGNE 1206 Introduction Torque curves are a common infographic given when you buy a DC motor. They tell you the capabilities of the motor, and the electrical requirements of the motor at a given motor torque. In this lab you will be given a data set to generate torque curves for a VEX 393 DC Motor. These curves include Torque-Angular Velocity, Torque-Current, Torque-Power, and Torque-Efficiency. The 393 motor is being tested on a crane device where it lifts a bucket filled with different masses. But before you begin the lab, you should know the mechanics of a DC motor, and the mechanics of the crane mechanism shown in the accompanying lab video. Methods Methods are as shown in the accompanying lab video. Theory (also in video) 1 - Definitions Angular velocity: The speed at which an object rotates. Measured in radians per second (rad/s). Torque: A force exerted at a distance from the center of rotation. Measured in Newton-meters (Nm). Voltage: The amount of energy needed to move an electric charge between 2 points. Measured in Volts (V). Current: The speed at which an electric charge moves between 2 points. Measured in Amperes (A). Power: The rate of energy transfer. Power has many forms, like energy. This lab focuses on mechanical and electrical power. Measured in Watts (W). Efficiency: The ratio between power put into a system vs. what comes out of the system. In this lab, the motor is the system, electrical power is the power put into the system (to power the motor), and mechanical power is the power coming out of the system (to lift the bucket). It is unitless and typically expressed as a percentage. 2 - DC Motors DC motors in free spin (no load on the motor, just a spinning shaft) have 2 linear relationships that they follow: The angular velocity of the motor is proportional to the voltage supplied to the motor. The torque of the motor is proportional to the current supplied to the motor. But when a motor is under load, the relationship between angular velocity and voltage breaks down. The angular velocity will begin to slow as more torque is applied, despite the voltage staying the same. The relationship between torque and current remains true. The more torque that is applied, the more current the motor needs to resist it. All motors have their limits. DC motors often come with a label or datasheet that tells you the stall torque. This is the maximum torque that a motor can withstand before it can no longer run. 3 - Gear Ratios When 2 gears on different axes are introduced to a mechanical system, they change the value of the angular velocity and torque. This is because they must maintain the same velocity and force where the gears meet. Each tooth of a gear pushes the next one forward, and this 1:1 tooth ratio is the reason that they travel at the same velocity. Because velocity is the same, acceleration is 0, and thus there is no difference in net force where the gears mesh. However, the radius of each gear is different. This means that the angular velocity is different (think: the larger gear has more circumference, so it will take longer to rotate completely compared to the smaller gear with a smaller circumference.), and the torque is different (the bigger gear has a larger radius, which means that the force between them creates a larger torque in the large gear than the small gear). 2 1 Remember that velocity equals angular velocity times radius. Because the velocity is the same at the edge of each gear, the angular velocity and radius of each gear can be related to each other. V = w*r V1 = V2 W1*r1 = w2*r2 A similar yet different equivalency exists between torque and radius. Torque is equal to the force between gears times a gear radius. Because the force is equal, torque changes depending on the radius of each gear as well. T = F*r F = T/r F1 = F2 T1/r1 = T2/r2 These relationships also exist when 2 gears are on the same axis, but slightly different. When the gears are on the same axis, they rotate together. This means that their angular velocity is the same. But because the radii are different, the have different velocities at the edge of the gears. V = w*r w = V/r w1 = w2 V1/r1 = V2/r2 When you look at torque and force of 2 gears on the same axis, torque stays constant for both gears. This is because if the torques were unequal, the shaft would twist and break (and if you watch the video, you will see that it does not). Torque is constant, so the force on the edge of each gear differs with the radius of the gear. T = F*r T1 = T2 F1*r1 = F2*r2 Using the 4 relationships above, equations to find the angular velocity and torque exerted on the motor and be found, as functions of the weight and velocity of the bucket. These equations are below: Tm = Fc*rc*ra/rb (1) Where: Tm = Torque exerted on the motor Fc = Weight of the bucket rc = Radius of the fishing wire spool ra = Radius of the lower gear rb = Radius of the upper gear wm = (Vc * rb) / (ra * rc)(2) Where: wm = Angular velocity of the motor Vc = Velocity of the bucket 4 - Power and Efficiency Power is put into the system electrically by supplying voltage and current to the motor. The electrical power is calculated by multiplying the voltage and the current together. Pin = Pelec = V * I(3) Power is taken out of the system to lift the bucket. The mechanical power is calculated by multiplying the torque and the angular velocity of the motor. Pout = Pmech = T * w(4) In the real world, there are always losses of energy through friction, heat transfer, electrical impedence, as well as many other factors. Because of this, it is impossible to get 100% of the power put into a system to come out in the form you want. Efficiency tells us how much power is being used for what it is intended for, and how much is lost through friction, heat, impedence, etc. Efficiency is found by dividing the mechanical power by the electrical power and multiplying by 100%. N = (Pout / Pin) * 100(5) Results 1. Fill in the rest of the calculations using equations 1-5 from the Theory section. The first 2 horizontal lines of data should match the picture below if your formulas are entered correctly. 2. Create scatter plots of Torque vs. Angular Velocity and Torque vs. Current. Make sure the axes are labelled with proper units. Apply a line of best fit to both. Write the equation of the line of best fit and the R2 value below. Torque vs. Angular Velocity Torque vs. Current Equation: Equation: R2: R2: 3. Create a scatter plot of Torque vs. Power. Make sure the axes are labelled with proper units. Make sure to include both Pin and Pout on the plot. Apply lines of best fit to each series and write the equation and R2 value below. Torque vs. Power In Torque vs. Power Out Equation: Equation: R2: R2: 4. Add the calculated efficiency to the Torque vs. Efficiency plot. Submit your completed spreadsheet with this assignment. Discussion 1. What similarities do you notice between the lines of best fit for Angular Velocity vs. Torque and Pout vs. Torque? 2. Does it make sense that Pout is not linear when plotted against torque? Why? 3. What torque gives the highest ideal efficiency? Why would it be useful to know this information? Give a real-life example. Results Section #1ConstantsRaw DataCalculations ra (m)0.045Mass (kg)Time (s)Current (A)Fc (N)Vc (m/s)Tm (Nm)wm (rad/s)Pin (W)Pout (W)Efficiency (-) rb (m)0.0321.20313.151.0311.801430.0866920152 rc (m)0.0151.10110.350.9510.800810.1101449275 Supplied voltage (V)7.10.9989.350.99.790380.1219251337 Height of table (m)1.140.9038.450.848.858430.1349112426 gravity acceleration (m/s^2)9.810.7997.720.797.838190.1476683938 0.7037.060.736.896430.1614730878 0.5976.140.665.856570.1856677524 0.5025.820.584.924620.1958762887 0.4015.50.523.933810.2072727273 0.35.040.482.9430.2261904762 0.1994.840.41.952190.2355371901 0.1674.260.41.638270.2676056338 #2 (Put graphs here) #3 (put graph here) #4 (put ideal efficiency on this graph) Efficiency of a Motor Under Various Loads Ideal Efficiency5.0000000000000001E-30.011.4999999999999999E-20.022.5000000000000001E-20.033.5000000000000003E-20.044.4999999999999998E-20.055.5E-20.066.5000000000000002E-27.0000000000000007E-27.4999999999999997E-20.088.5000000000000006E-20.099.5000000000000001E-20.10.1050.110.1150.120.1250.130.135000000000000010.140000000000000010.144999999999999990.150.1550.160.165000000000000010.170.174999999999999990.180.1850.190.195000000000000010.20.204999999999999990.210.2150.220.225000000000000010.230.234999999999999990.240.2450.250.2550.260.265000000000000010.270.275000000000000020.280000000000000030.284999999999999980.289999999999999980.294999999999999980.30.304999999999999990.310.3150.320.325000000000000010.330.335000000000000020.340.344999999999999970.350.354999999999999980.360.364999999999999993.0068899646969265.65277364587374017.986085995644718810.04696840758637911.868970987858413.48035208515899114.9050820278738116.16362674564173617.27356560517317918.25008298652538119.10636271079620819.85390700399760920.50279632103487521.06190243913962221.53906434087315721.94123425296091822.27459958649160822.54468529371679322.7564402150383122.91431026343713723.02230072922979823.08402954647726123.10277301462570123.08150519339209723.0229319692492522.92952061578936122.80352552835141122.64701069836835522.46186939934969122.24984147992972122.0125285965955821.75140766690062621.46784278106467221.16309577419594120.83833563160693820.49464687477308520.13303705453936119.75444346052245219.35973914072151218.94973831268059418.52520123676453818.08683861291073217.63531555434694917.17125518501179116.69524190160483816.20782433618425615.70951805089960415.20080799269585714.68215073256754314.15397651110638613.61669110961418913.07067756389174912.51629773592311811.95389375701474511.38378935449022910.80629107275692210.2216893984273569.63025979817714279.03226367713475048.42794926481258957.8175524348911647.20129746454740266.57939773846535395.95205640217456195.31946696892067464.68181388387897184.03927304916829673.39201231280654452.74019192446413932.08396496061647121.4234777214658210.758870101795826269.027593773411513E-2Torque (Nm) Efficiency (-) Ideal Efficiency (don't touch!) Torque (Nm)Ideal Efficiency (-) 0.0053.0068899647 0.015.6527736459 0.0157.9860859956 0.0210.0469684076 0.02511.8689709879 0.0313.4803520852 0.03514.9050820279 0.0416.1636267456 0.04517.2735656052 0.0518.2500829865 0.05519.1063627108 0.0619.853907004 0.06520.502796321 0.0721.0619024391 0.07521.5390643409 0.0821.941234253 0.08522.2745995865 0.0922.5446852937 0.09522.756440215 0.122.9143102634 0.10523.0223007292 0.1123.0840295465 0.11523.1027730146 0.1223.0815051934 0.12523.0229319692 0.1322.9295206158 0.13522.8035255284 0.1422.6470106984 0.14522.4618693993 0.1522.2498414799 0.15522.0125285966 0.1621.7514076669 0.16521.4678427811 0.1721.1630957742 0.17520.8383356316 0.1820.4946468748 0.18520.1330370545 0.1919.7544434605 0.19519.3597391407 0.218.9497383127 0.20518.5252012368 0.2118.0868386129 0.21517.6353155543 0